Question

I want to compute y = a⊗a⊗a, where a is a n-by-1 vector, and ⊗ is the outer product operator. In this case y should be an n-by-n-by-n tensor.

If y = a⊗a, it is easy. I simply do:

y = a * a' 

But what to do in the first case? How do I compute this outer product efficiently in MATLAB if there are more than two vectors?

Answer 1

In a multi-dimensional (tensor) case of y = u⊗v, I believe that you need to shift the dimensions of the second operand like so:

v_t = permute(v, circshift(1:(ndims(u) + ndims(v)), [0, ndims(u)]));

and then multiply them with bsxfun:

y = bsxfun(@times, u, v_t);

The regular matrix multiplication is defined only for vector and 2-D matrices, so we couldn’t use it in the general case.

Also note that this computation still fails if the second operand is a 1-D vector, because ndims returns 2 instead of 1 for vectors. For this purpose, lets define our own function that counts dimensions:

my_ndims = @(x)(isvector(x) + ~isvector(x) * ndims(x));

To complete the answer, you can define a new function (e.g. an anonymous function), like so:

outprod = @(u, v)bsxfun(@times, u, permute(v, circshift(1:(my_ndims(u) + my_ndims(v)), [0, my_ndims(u)])));

and then use it as many times as you want. For example, y = a×a×a would be computed like so:

y = outprod(outprod(a, a), a);

Of course, you can write a better function that takes a variable number of arguments to save you some typing. Something along these lines:

function y = outprod(u, varargin)
    my_ndims = @(x)(isvector(x) + ~isvector(x) * ndims(x));
    y = u;
    for k = 1:numel(varargin)
        v = varargin{k};
        v_t = permute(v, circshift(1:(my_ndims(y) + my_ndims(v)),[0, my_ndims(y)]));
        y = bsxfun(@times, y, v_t);
    end

I hope I got the math right!